The first sweet spot is named the Center of
Percussion. If you swing a bat and hit a ball near the end of the
bat, the bat will pivot about a point near its Center of Mass and the
handle will push your hands forward. If on the other hand you hit a
bat at its Center of Mass the entire bat will try to move backward
without rotating pushing your hands backward. In between these two
positions of the bat is the center of percussion where the forward
motion of the handle exactly cancels the backward motion of the
center of mass.

The second sweet spot is the node of vibration.
Hit a bat near the fat end and it will vibrate and even make a tone
you can hear. Strike the bat at different points moving slowly up the
bat and you will find a point where the bat does not vibrate much at
all. The point where the bat does not vibrate much when you strike it
is the node of vibration.